On necessary conditions for Fourier multipliers of weak type

We obtain necessary conditions for a bounded function $\varphi$ to be a Fourier multiplier of weak type $\bigl(\psi(L)(G),\psi(L)(G)\bigr)$, where $G=\mathbb R^d$ or $G=\mathbb T^d$, provided that the Young function $\psi(t)$ grows slower than $t\ln^{1/2}t$ as $t$ tends to infinity.